Minimal Length Scale Scenarios for Quantum Gravity

Abstract

We review the question of whether the fundamental laws of nature limit our ability to probe arbitrarily short distances. First, we examine what insights can be gained from thought experiments for probes of shortest distances, and summarize what can be learned from different approaches to a theory of quantum gravity. Then we discuss some models that have been developed to implement a minimal length scale in quantum mechanics and quantum field theory. These models have entered the literature as the generalized uncertainty principle or the modified dispersion relation, and have allowed the study of the effects of a minimal length scale in quantum mechanics, quantum electrodynamics, thermodynamics, black-hole physics and cosmology. Finally, we touch upon the question of ways to circumvent the manifestation of a minimal length scale in short-distance physics.

Keywords: generalized uncertainty principle; minimal length; quantum gravity.

Figure 1

Heisenberg’s microscope. A photon moving along the x-axis scatters off a probe within an interaction region of radius R and is detected by a microscope (indicated by a lens and screen) with opening angle ϵ.

Figure 2

The length of a string is not the same as its average extension. The lengths of strings in the groundstate were studied in [173].

Figure 3

Spacetime uncertainty (red, solid) vs uncertainty from spherical black holes (blue, dotted) in D = 10 dimensions, for g_s < 1 (left) and g_s > 1 (right). After [318], Figure 1. Below the bound from spacetime uncertainty yet above the black-hole bound that hides short-distance physics (shaded region), the concept of classical geometry becomes meaningless.

Figure 4

Cross section for scattering of two scalar particles by graviton exchange with and without running Planck mass, in units of the low-energy Planck mass . The dot-dashed (purple) line depicts the case without asymptotic safety; the continuous (blue) and dashed (grey) line take into account the running of the Planck mass, for two different values of the fixed point, and 0.1 respectively. Figure from [261]; reproduced with permission from IOP.